In a large pot, heat the olive oil over medium heat. Add the chopped onion, carrot, and celery. Sauté for about 5 minutes until the vegetables begin to soften.
Stir in the minced garlic and cook for another minute until fragrant.
Add the diced potatoes, vegetable or chicken broth, Cajun seasoning, and smoked paprika to the pot. Bring the mixture to a boil.
Once boiling, reduce the heat to low, cover, and let it simmer for 20-25 minutes, or until the potatoes are tender.
Using an immersion blender, carefully blend the soup until smooth. If you don’t have an immersion blender, transfer the soup in batches to a countertop blender.
Stir in the heavy cream and season with salt and pepper. Let it simmer for an additional 5 minutes to heat through.
Adjust seasoning if necessary and remove from heat.
Serve hot, garnished with chopped green onions and fresh parsley.